Area and Circumference of a Circle by
1. Understand the relationship among , area, and circumference of a
2. Understand how the unit circle (radius = 1) and the concept of
mathematical limits was used by Archimedes to show that and
1. The area of a polygon with n sides (n-gon) inscribed in a unit circle
approaches the special number pi as n increases.
2. The area of an inscribed polygon approaches the area of a circle as
the number of sides on the polygon increases.
3. The perimeter of an inscribed polygon approaches the perimeter
(circumference) of a circle as the number of sides on the polygon increases.
Knowns and assumptions
- n = number of sides on the inscribed circle
- 1 = radius of unit circle
- = height of an isosceles triangle inscribed in the inner circle
- = base of an isosceles triangle inscribed in the inner circle
- R = radius of the n-gon (Note that this radius is visualized in this applet
as being greater than one, but R could be any value greater than
- = height of each inscribed isosceles triangle (based on = 1)
- = base of each inscribed isosceles triangle (based on = 1)
- = the perimeter of the unit circle
- = area of the unit circle
- A = the sum of the areas of the inscribed triangles, the limit of which is
the area of the circle
- C = the sum of the "bases" of the inscribed isosceles triangles, the limit
of which is the circumference of the circle.
- Area of a triangle =
- Other basic rules of algebra and geometry
With the unit circle (red) as the basis, Archimedes used the limiting
process on the area
and base of polygons (n-gons) inscribed in circles (as n approaches
infinity) to determine and at the same time verify the formulas for the
area and circumference of any circle (blue).
What to do
- Notice how R, , and
the unit circle are used in the graphical Definition of Variables.
- Observe how each of these variables are positioned in the equation in
the center bottom of the screen.
- Follow how the rules of algebra are used to rearrange the variables from
the center equation outward to the left and outward to the right. Also,
use of in the right-hand limit.
- Increase n. Note that as you increase n, the values of on the
left and the value of on the right side simultaneously come closer
and closer to the limiting value .
- Notice: The limit at the left is A, the area of the circle. The limit
at the right is , the perimeter (or circumference) of the circle.
Click the Zoom In and Zoom Out to toggle among 1:1, 16:1,
and 32:1 zoom levels to see close-ups around a point on the outer circle
for which the area and perimeter are being approximated. Zoom lets you
visualize that the perimeter of the n-gon gets closer and closer to the
perimeter of the circle, and the area of the n-gon gets closer and closer
to the area of the circle.
- From the left-hand limit, you see that
- On the right side you get the limit .
- Since these limits are occurring simultaneously, =
- By solving for C in the equation = ,
it follows that .